Area Calculator | 11 Shapes with Formulas & Perimeter
Calculate area and perimeter for 11 shapes: rectangle, square, circle, triangle, right triangle, trapezoid, parallelogram, rhombus, ellipse, hexagon, and sector. Formulas shown with your values substituted in.
What Is the Area Calculator | 11 Shapes with Formulas & Perimeter?
This area calculator supports 11 common 2D shapes, covering everything you need for geometry homework, home improvement projects, landscaping, and engineering work. Beyond just area, each shape also shows its perimeter (or circumference) and any shape-specific properties like diagonal, eccentricity, or arc length.
- ▸11 shapes: Rectangle, Square, Circle, Triangle, Right Triangle, Trapezoid, Parallelogram, Rhombus, Ellipse, Regular Hexagon, and Circle Sector.
- ▸Perimeter alongside area: Every shape shows perimeter/circumference in the same calculation.
- ▸Shape-specific extras: Diagonal for rectangles and squares; hypotenuse and angles for right triangles; chord and arc for sectors; eccentricity for ellipses; apothem for hexagons.
- ▸Optional unit selector: Choose mm, cm, m, km, in, ft, or yd, the unit label is shown with area (unit²) and perimeter (unit) results.
- ▸Formula shown: The formula with your numbers substituted is displayed below every result, making it easy to verify and learn.
Formula
Rectangle
A = w × h
P = 2(w + h)
Square
A = s²
P = 4s, d = s√2
Circle
A = πr²
C = 2πr
Triangle
A = ½ × b × h
Height must be perpendicular to base
Right Triangle
A = ½ × a × b
c = √(a² + b²) (Pythagorean theorem)
Trapezoid
A = ½(a + b) × h
a, b = parallel sides; h = height
Parallelogram
A = b × h
P = 2(b + s), h ⊥ b
Rhombus
A = ½ × d₁ × d₂
Side = √((d₁/2)² + (d₂/2)²)
Ellipse
A = π × a × b
Ramanujan approx for perimeter
Regular Hexagon
A = (3√3/2) × s²
P = 6s, apothem = s√3/2
Circle Sector
A = ½r²θ (θ in rad)
Arc = rθ, chord = 2r sin(θ/2)
How to Use
- 1
Select the shape
Click the shape you need from the grid of icons at the top. The input fields update automatically to show only the measurements that shape requires.
- 2
Choose a unit (optional)
Pick your measurement unit from the dropdown (cm, m, ft, etc.). Area will be shown in square units, perimeter in linear units.
- 3
Enter the dimensions
Fill in all required fields with positive numbers. Each field is labelled clearly, width, height, radius, base, diagonal, etc.
- 4
Click Calculate Area
Results appear immediately showing area, perimeter, and any extra measurements specific to that shape.
- 5
Read the formula
The formula panel shows the equation used with your numbers substituted in, useful for checking your work or understanding how the result was obtained.
Example Calculation
Example 1 | Rectangle room (painting)
A room measures 5.5 m wide by 4.2 m long.
At 10 m² coverage per litre, you need about 2.31 litres of paint for one coat.
Example 2 | Circle lawn (sprinkler coverage)
A circular sprinkler covers a radius of 3 metres.
For a lawn of 100 m², you would need approximately 4 such sprinklers without overlap.
Example 3 | Right triangle plot
A triangular land plot has legs of 12 m and 9 m.
This is a 3-4-5 Pythagorean triple (scaled by 3): 9-12-15.
Example 4 | Hexagonal tile
A regular hexagonal floor tile has side length 15 cm.
Understanding Area | 11 Shapes with Formulas & Perimeter
Why Area Calculation Matters
Area calculations appear in almost every aspect of life, from figuring out how much carpet you need for a room, to designing a garden bed, to solving engineering and architecture problems. Getting the formula right matters: a 10% measurement error squares to a 21% area error, not just 10%. Understanding the geometry behind each formula helps you avoid these mistakes.
All 11 Shape Formulas, Explained
| Shape | Area Formula | Key Insight |
|---|---|---|
| Rectangle | w × h | Length × width; forms the basis for all other area formulas |
| Square | s² | Special case of rectangle with w = h |
| Circle | πr² | π is the ratio of circumference to diameter; r² scales with size |
| Triangle | ½ × b × h | A triangle is half a rectangle of the same base and height |
| Right Triangle | ½ × a × b | The two legs act as base and height since they meet at 90° |
| Trapezoid | ½(a+b) × h | Average of the two parallel sides times height |
| Parallelogram | b × h | Same as rectangle: the slant does not affect area, only height does |
| Rhombus | ½ × d₁ × d₂ | Diagonals bisect at right angles forming four right triangles |
| Ellipse | π × a × b | Generalises the circle formula: when a = b = r, A = πr² |
| Regular Hexagon | (3√3/2) × s² | Made of 6 equilateral triangles each with area (√3/4)s² |
| Sector | ½r²θ (θ in rad) | Fraction θ/2π of the full circle area πr² |
Converting Between Area Units
Area units scale by the square of the length unit. This trips up many people:
- ▸1 m² = 10,000 cm² (not 100, because both dimensions scale by 100)
- ▸1 km² = 1,000,000 m²
- ▸1 ft² = 144 in² (12 × 12)
- ▸1 acre = 4,047 m² = 43,560 ft²
- ▸1 hectare = 10,000 m² = 2.471 acres
Practical Applications by Shape
- ▸Rectangle: Floor space, wall area for painting, window glass, carpet, tile coverage, land parcels.
- ▸Circle: Pipe cross-sections, circular pools and ponds, pizza size, wheel and gear design.
- ▸Triangle: Roof sections, road signs, architectural gables, triangular plots of land.
- ▸Trapezoid: Drainage channels, road cross-sections, embankments.
- ▸Ellipse: Elliptical tables and mirrors, orbital cross-sections, optics.
- ▸Hexagon: Honeycomb structures, hexagonal floor tiles, nut and bolt head sizing.
- ▸Sector: Pizza slices, fan blade areas, rotational machine components, clock sector calculations.
Tips for Accurate Measurements
- ▸For irregular rooms, divide into rectangles and triangles. Measure each part separately and add the areas.
- ▸Always measure height perpendicular to the base, not the slant height, for triangles, parallelograms, and trapezoids.
- ▸For circles, it is easier to measure diameter than radius. Enter half the diameter into the radius field.
- ▸Add 10–15% to material estimates for waste, cuts, and overlap when tiling or laying flooring.
- ▸For complex curved shapes, use the area between curves calculator with a mathematical function that approximates the boundary.
Frequently Asked Questions
What units does this calculator use?
The calculator works with any consistent unit you choose. If you enter measurements in metres, the area result is in square metres (m²) and perimeter in metres. Select your unit from the dropdown to have the labels displayed correctly.
What is the difference between area and perimeter?
Area measures the surface enclosed within a shape (square units). Perimeter measures the total distance around the boundary (linear units). For example, a 4 × 3 rectangle has area = 12 m² but perimeter = 14 m.
How do I find the area of an irregular shape?
Break it into simpler shapes (rectangles, triangles, circles), calculate each area separately, and add them together. For concave cut-outs, subtract the cut-out area from the outer shape.
Why does a rhombus use diagonals instead of sides?
A rhombus has four equal sides, but that alone does not determine the area, it depends on the angles (or equivalently, the diagonal lengths). The formula A = ½d₁d₂ works because the diagonals of a rhombus bisect each other at right angles, forming four right triangles.
Why is the ellipse perimeter formula approximate?
Unlike other shapes, there is no simple exact formula for the perimeter of an ellipse. This calculator uses Ramanujan's second approximation: P ≈ π(a+b)[1 + 3h/(10+√(4−3h))] where h = ((a−b)/(a+b))². It is accurate to within 0.0003% for most eccentricities.
What is the apothem of a hexagon?
The apothem is the perpendicular distance from the centre of a regular polygon to the middle of one of its sides. For a regular hexagon with side length s, the apothem = s√3/2. It is also the inradius, the radius of the inscribed circle that touches all six sides.
How is sector area different from segment area?
A sector is the pie-slice region bounded by two radii and the arc (like a slice of pizza). A segment is the region between the chord and the arc (just the crust, without the triangular part). Sector area = ½r²θ; segment area = sector area − triangle area = ½r²(θ − sinθ).